Τοπολογική Τετριμμενοποίηση
Τετριμμενοποίησις Trivialization thumb|300px| [[Τοπολογική Τετριμμενοποίηση |Τετριμμενοποίηση ]] thumb|300px| [[Τοπολογική Τετριμμενοποίηση |Τετριμμενοποίηση ]] thumb|300px| [[Τοπολογική Τετριμμενοποίηση |Τετριμμενοποίηση ]] thumb|300px| [[Τοπολογική Τετριμμενοποίηση |Τετριμμενοποίηση ---- In general, tangent and cotangent bundles, T M and T ∗ M , of a smooth manifold M, are special cases of a more general geometrical object called "fibre bundle", denoted π : Y → X, where the fiber V of a map π : Y → X is the preimage π − 1 ( x ) of an element x ∈ X. It is a space which locally looks like a product of two spaces (similarly as a manifold locally looks like Euclidean space), but may possess a different global structure. To get a visual intuition behind this fundamental geometrical concept, we can say that a fibre bundle Y is a homeomorphic generalization of a product space X × V (see Figure 2), where X and V are called the base and the fibre, respectively. π : Y → X is called the projection , Y x = π − 1 ( x ) denotes a fibre over a point x of the base X, while the map f = π − 1 : X → Y defines the cross–section, producing the graph (x, f(x)) in the bundle Y (e.g., in case of a tangent bundle, f = x ̇ represents a velocity vector–field). ]] thumb|300px| [[Τετριμμενότητα ]] - Μία διαδικασία. Ετυμολογία Η ονομασία "Τετριμμενοποίηση" σχετίζεται ετυμολογικά με την λέξη "τριβή". Εισαγωγή In mathematics, and particularly topology, a fiber bundle is a space that is locally a product space, but globally may have a different topological structure. Specifically, the similarity between a space E'' and a product space ''B × F'' is defined using a continuous surjective map : \pi\colon E \to B that in small regions of ''E behaves just like a projection from corresponding regions of B'' × ''F to B''. The map π, called the 'projection' or 'submersion' of the bundle, is regarded as part of the structure of the bundle. The space ''E is known as the total space of the fiber bundle, B'' as the '''base space', and F'' the '''fiber'. In the trivial case, E'' is just ''B × F'', and the map π is just the projection from the product space to the first factor. This is called a ''trivial bundle. Πλέον φορμαλιστικά: A real vector bundle consists of: # topological spaces X'' (''base space) and E'' (''total space) # a continuous surjection π : E'' → ''X (bundle projection) # for every x'' in ''X, the structure of a finite-dimensional real vector space on the fiber π−1({x''}) where the following compatibility condition is satisfied: for every point in ''X, there is an open neighborhood U'', a natural number ''k, and a homeomorphism : \varphi\colon U \times \mathbf{R}^{k} \to \pi^{-1}(U) such that for all x'' ∈ ''U, * (\pi \circ \varphi)(x,v) = x for all vectors v'' in '''Rk'', and * the map v \mapsto \varphi (x, v) is a linear isomorphism between the vector spaces '''Rk'' and π−1({''x}). The open neighborhood U'' together with the homeomorphism φ is called a '''local trivialization' of the vector bundle. The local trivialization shows that locally the map π "looks like" the projection of U'' × '''R'k on U. Υποσημειώσεις Εσωτερική Αρθρογραφία * Τοπολογική Γειτονία * Ανοικτό Σύνολο * Τοπολογική Συνάρτηση * Ομοιομορφισμός * Τοπολογικός Άτλας * Τοπολογικός Χάρτης Βιβλιογραφία * * Ιστογραφία * Ομώνυμο άρθρο στην Βικιπαίδεια * Ομώνυμο άρθρο στην Livepedia * mathworld.wolfram.com Κατηγορία:Τοπολογία